We are concerned with a two-phase fluid model in \({\mathbb {R}}^3\). This model was first derived by Choi (SIAM J. Math. Anal. 48: 3090–3122, 2016) by taking the hydrodynamic limit from the Vlasov–Fokker–Planck/isentropic Navier–Stokes equations with strong local alignment forces. Under the assumption that the \(H^3\) norm of the initial data is small but its higher-order Sobolev norm can be arbitrarily large, the global existence and uniqueness of classical solutions are obtained by an energy method. Moreover, if in addition, the initial data norm of the \(\dot{H}^{-s}(0\le s<\frac{3}{2})\) or \(\dot{B}^{-s}_{2,\infty }(0< s\le \frac{3}{2})\) is small, we also obtain the optimal time decay rates of solutions.

我们关注\({\mathbb {R}}^3\) 中的两相流体模型。该模型首先由 Choi (SIAM J. Math. Anal. 48: 3090–3122, 2016) 通过从 Vlasov-Fokker-Planck/等熵 Navier-Stokes 方程中获得流体动力学极限而推导出来，并具有很强的局部对齐力。在假设初始数据的\(H^3\)范数很小但其高阶Sobolev范数可以任意大的情况下，通过能量方法获得经典解的全局存在唯一性。此外，如果另外，\(\dot{H}^{-s}(0\le s<\frac{3}{2})\)或\(\dot{B}^ {-s}_{2,\infty }(0< s\le \frac{3}{2})\)很小，我们也得到了最优解的时间衰减率。